Cascaded Eddy Simulation

ABSTRACT

A fluid flow is simulated by causing a computer to perform operations on data stored in the memory to compute at least one eddy of a fluid flow at a first scale and perform operations to compute at least one eddy of the fluid flow at both the first scale and a second scale. The second scale is a finer scale than the first scale, and the computation of the at least one eddy of the fluid flow at the second scale is constrained by results of the computation of the at least one eddy of the fluid flow at the first scale.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.60/830,805, filed Jul. 14, 2006, and titled CASCADED EDDY SIMULATION,which is incorporated by reference in its entirety.

TECHNICAL FIELD

This description relates to computer systems for simulating physicalprocesses, e.g., a fluid flow.

BACKGROUND

A fluid may be generally defined as any substance that can flow. Fluidsencompass gases, liquids and combinations of gases and liquids. Thefield of fluid dynamics attempts to explain and characterize thebehavior of fluids. The behavior of fluids may be characterized throughdifferential equations.

SUMMARY

In one general aspect, a computer simulates a fluid flow by performingoperations on data stored in the memory to compute at least one eddy ofa fluid flow at a first scale and performing operations to compute atleast one eddy of the fluid flow at both the first and a second scale.The second scale is a finer scale than the first scale, and thecomputation of the at least one eddy of the fluid flow at the secondscale is constrained by results of the computation of the at least oneeddy of the fluid flow at the first scale.

Implementations may include one or more of the following features. Forexample, the computation of the at least one eddy of the fluid flow atthe combined first and second scales may be constrained such that anenergy spectrum of the fluid flow is substantially the samestatistically as the results of the computation of the at least one eddyof the fluid flow at the first scale.

Operations may be performed to compute at least one eddy of the fluidflow at a third scale where third scale is a finer scale than the secondscale, and the computation of the at least one eddy of the fluid flow atthe third scale is constrained by results of the computation of the atleast one eddy of the fluid flow at the second scale.

A region of the fluid flow at the third scale may be identified, wherethe region represents a region of activity that is significant to thecomputing of the fluid flow at the third scale, and operations may beperformed to compute at least one eddy of the identified region of thefluid flow at a third scale. Operations may be performed toconditionally sample the fluid flow at the third scale to identify theregion of significant activity. Conditionally sampling the fluid flow atthe third scale may include conditionally sampling at least one of localshear, local vorticity and a combination of shear and vorticity or someother flow quantity.

Operations to compute at least one eddy of the fluid flow at a secondscale may include determining a modification of the flow equation usedto compute the at least one eddy of the fluid flow at the second scale,and computing the at least one eddy of the fluid flow at the secondscale using the flow equation as modified.

Operations to compute at least one eddy of the fluid flow at a secondscale further may include analyzing results of the computation at thesecond scale to determine whether the forcing is to be modified. Inresponse to a determination that the forcing is to be modified, adetermination may be made whether a second forcing is to be applied by aflow equation used to compute the at least one eddy of the fluid flow atthe second scale. The computing the at least one eddy of the fluid flowat the second scale may use the flow equation modified by the determinedsecond forcing. The modified flow equation used to compute the at leastone eddy of the fluid flow at the second scale may include a modifiedNavier-Stokes equation, a modified kinetic equation, a modified flowequation that includes a relaxation term, or a modified flow equationthat includes a dampening term.

In another general aspect, a computer system for simulating a fluid flowincludes a processor, a memory and a mass storage device. The system isconfigured to perform operations on data stored in the memory to computeat least one eddy of a fluid flow at a first scale and performoperations to compute at least one eddy of the fluid flow at a secondscale. The second scale is a finer scale than the first scale, and thecomputation of the at least one eddy of the fluid flow at the secondscale is constrained by results of the computation of the at least oneeddy of the fluid flow at the first scale.

Implementations may include one or more of the features noted above andone or more of the following features. For example, the system may beconfigured to perform operations only on data stored in the memory tocompute at least one eddy of the fluid flow at the second scale. Thesystem may be configured to perform operations on data stored in thememory and data stored on the mass storage device to compute at leastone eddy of the fluid flow at the second scale.

Implementations of the techniques discussed above may include a methodor process, a system or apparatus, or computer software on acomputer-accessible medium.

Other features will be apparent from the following description,including the drawings, and the claims.

DESCRIPTION OF DRAWINGS

FIGS. 1A-1C are diagrams of a simulation of a turbulent flow over atangible object.

FIG. 2 is a block diagram of a cascade of eddy sizes in a turbulentflow.

FIG. 3 is a diagram illustrating the qualitative energy spectrum of aturbulent flow as energy is dissipated from coarse eddies to dissipativescale eddies.

FIG. 4 is a flow chart of an example process for a cascaded eddysimulation.

FIG. 5 is a flow chart of an example process for computing somecoarse-scale eddies of a turbulent flow.

FIG. 6 is a flow chart of an example process 600 for computing cascadededdies resulting from the coarse-scale eddies of a turbulent flow.

FIGS. 7A and 7B are diagrams of qualitative energy spectrums of aturbulent flow.

FIG. 8 is a block diagram of a computer system capable of performing acascaded eddy simulation.

FIG. 9 is a flow chart of an example process for computing the turbulentflow of finer-scale eddies.

DETAILED DESCRIPTION

Turbulent flows are characterized by a large range of scales of motion.Indeed, the general theory of turbulence characterizes turbulent flowsas having a range of excitations ranging from coarse eddies,representative of the scales at which the turbulence is formed eitherdue to forcing or geometry, to fine eddies which act to dissipate theenergy input at the coarser scales. Thus, one view of turbulent flowsinvolves a “cascade” of eddy sizes as well as a “cascade” of energy fromthe coarse, energy-containing eddies to fine-scale “dissipating eddies.”

FIGS. 1A-1C illustrate a simulation of a turbulent flow 100 over atangible object 110 (here, a step). The turbulent flow 100 is passingfrom the point 115 to the point 120 in FIGS. 1A-1C.

FIG. 1A illustrates the averaged flow 100A over a time period from 31seconds to 61 seconds from the start of the simulation. As such, theaveraged flow 100A illustrates an example of a coarse scale eddy of theflow, such as coarse scale eddy 210 described below with respect to FIG.2. The averaged flow 100A includes coarse eddies 125A and 130A, as wellas a portion 135A of the averaged flow that does not appear to include acoarse eddy. FIGS. 1B and 1C each depict examples of finer scale eddiesthat form within the flow 100 at particular points of time.

More particularly, FIG. 1B depicts a snapshot of the flow 100 at about31 seconds from the start of the simulation and shows finer scale eddies125B and 130B that form within the coarse scale eddies 125A and 130A atthat point of time, as well as a finer scale eddy 135B that forms in theportion 135A. The finer scale eddies 125B, 130B and 135B are examples offine scale eddies of the turbulent flow 100, such as finer scale eddies230 described below with respect to FIG. 2. As illustrated, the finerscale eddies 125B show more detail than the corresponding coarse scaleeddy 125A of FIG. 1A. Similarly, the finer scale eddy 130B shows moredetail than the coarse scale eddy 130A of FIG. 1A. In contrast with theportion 135A of the flow of FIG. 1A, the finer scale eddy 135B isvisible.

FIG. 1C shows a snapshot of the flow 100 at an even later time—that is,about 61 seconds from the start of the simulation—and illustrates stillfiner scale eddies that form at that later point in time within coarsescale eddies. The finer scale eddies 125C and 130C are examples of finerscale eddies of the turbulent flow 100, such as finer scale eddies 240described below with respect to FIG. 2. As illustrated, the eddies 125Cand 130C show more detail than the corresponding coarse scale eddies125A and 130A of FIG. 1A. In contrast with eddy 135B of FIG. 1B, a finerscale eddy 135C has nearly dissipated.

FIG. 2 conceptually illustrates a cascade 200 of eddy sizes 210-250 in aturbulent flow. In general, there is a cascade of energy input at acoarse scale to dissipation of the energy at the finest scales. Moreparticularly, and as described in more detail later with respect to FIG.3, there is a cascade of energy from coarse scale eddies 210 tointermediate scale eddies 220 to inertial scale eddies 230, 240 and todissipative scale eddies 250. There may be more scales of inertialeddies shown, which progressively get finer as energy flows from theintermediate eddies 220 to the dissipative eddies 250. In this example,inertial eddies 230 may be referred to as “fine inertial eddies” andinertial eddies 240 may be referred to as “finer inertial eddies.”

FIG. 3 illustrates the qualitative energy spectrum 300 of a turbulentflow as energy is dissipated from coarse eddies 210 to dissipative scaleeddies 250, where the dissipation of energy input at the coarsest scaleis completed. In particular, energy is input into the turbulent flow atthe coarsest-scale eddies 210, and energy is dissipated from theturbulent flow at the very finest scales—here, called dissipative scaleeddies 250.

The qualitative energy spectrum shown in FIG. 3 is consistent with theunderlying dynamics of flows, governed by the Navier-Stokes (N-S)equation, or its extensions to more complex flow situations such ascompressible or multiphase flows:

$\begin{matrix}{{\frac{\partial v}{\partial t} + {v \cdot {\nabla\; v}}} = {{- ( {1/\rho} )}{\nabla_{p}{+ v}}{{\nabla^{2}v}.}}} & (1)\end{matrix}$

Here v(x,t) is the velocity field which depends on the three-dimensionalposition x and the time t, ρ is the density, p is the pressure field,and v is the kinematic viscosity of the flow. The solutions to the N-Sequation are turbulent when v→0; indeed, it is conventional tonon-dimensionalize the N-S equation in terms of the Reynolds number R,which is defined as the non-dimensional quantity

R=UL/ν,

where U is a typical coarse scale velocity magnitude and L is the lengthscale over which this velocity U is coherent. When R is small, flows aretypically non-turbulent. Stated differently, flows are typically laminarwhen R is small, and, as R increases, flows typically first undergotransition from laminar to turbulence, and then, at coarse R, flowsbecome “fully turbulent.” When R is large, there is a significantchallenge in solving the N-S equation by numerical methods. Thechallenge is that, as R increases, the excited eddies of a turbulentflow have a range of sizes and times scales that increase at least atthe rate of R^(3/4) so the numerical resolution must increase rapidlywith R. With three space dimensions plus time, this result implies thatthe computational work scales as (R^(3/4))⁴=R³. The simulation of theturbulent flow past real-world coarse-scale flows (like flow past alarge ship) has been estimated to require on the order of Avogadro'snumber (6×10²³) or more arithmetic computations, which is well beyondthe capability of today's supercomputers. Thus, it is necessary to makeeither theoretical or numerical simplifications in the analysis of suchreal-world flows.

Techniques are provided for computationally simulating a turbulent flowby (1) computing some coarse-scale eddies of the turbulent flow and (2)computing the remaining scales to a resolution-limited grid scale toobtain cascaded eddies that are typically produced by the computedcoarse-scale eddies. The techniques may be implemented in a physicalprocess simulation system that uses cascaded eddy simulation (CES) toaccurately and effectively computationally simulating turbulent flows incomplex geometries at scales limited only by the available centralmemory of a computer system. Additional techniques are described thathelp to mitigate memory limitations.

In general, at least some of the coarse scale eddies of a turbulent floware computed to provide statistical limits, within which finer scaleeddies typically produced by the coarse scale eddies are computed. Whilespecifics of the coarse scale eddies may be modified in the computationof finer scale eddies, the statistical properties of the coarse scaleeddies remain largely intact in the presence of finer scale eddies. Thefiner scale eddies may be referred to as “cascaded eddies.” Computationsof eddies at finer scales may be iteratively performed to obtaincascaded eddies to finer scales where molecular dissipation iseffective. All of the cascaded eddies are limited by the statisticalproperties of the coarse scale eddies.

Referring to FIG. 4, a process 400 for a cascaded eddy simulation beginswith computation of coarse-scale eddies of interest in the specificturbulent flow (step 410). To do so, various methods, including VeryLarge Eddy Simulation, Large Eddy Simulation, Reynolds AveragedNavier-Stokes Simulation or another method are used to compute at leastsome of the coarser eddies of a turbulent flow, especially taking intoaccount the flow geometry, flow forcing and other details of the flowenvironment. Various numerical methods may be used, including, but notlimited to, finite difference, finite element, spectral and spectralelement solutions of suitable averaged Navier-Stokes equations orlattice Boltzmann or BGK kinetic methods with or without appropriateturbulence models. As an example of this step, lattice BGK tools (suchas PowerFlow code by Exa Corporation) may be used to compute thevery-coarse eddies of complex flows. These lattice BGK tools have theadded advantage that results are nearly independent of Reynolds numberwhen the Reynolds number is high and results typically lead to coarseeddies that exhibit some degree of time dependence, if appropriate. Moregenerally, step 410 may be referred to as the first step or step 1 ofthe CES process.

Next, there is a computation down to a resolution-limited grid scalethat is limited by the flow dynamics obtained in step 410 (step 420). Inother words, eddies of the turbulent flow are computed at a finer-scaleand the flow dynamics of the fine-scale eddies are constrained byresults of the coarse-scale eddy computation. In step 420, the flowdynamics obtained in step 410, which typically are limited in resolutionto fairly coarse eddies, are enhanced by a second computation. Thissecond computation is, in a general sense, controlled by the firstsimulation results so that the coarse eddies computed by step 410 arenot modified in a major way in the statistical sense. While details ofthe coarse eddies computed in step 410 may be modified in step 420, thestatistical properties of the coarse eddies remain largely intact. Forexample, boundary condition assumptions consistent with the coarse-scaleeddies may be applied to the computation of the finer-scale eddies. Thesecond computation in step 420 is to obtain the “cascaded eddies”typically produced by the coarse eddies of step 410 and that typicallycascade to finer scales where molecular dissipation is effective. Moregenerally, step 420 may be referred to as the second step or step 2 ofthe CES process.

The second computation (in step 420) may be done in several ways. In oneexample, the cascaded eddies are computed by solving the flow equations(whether the flow equations be the Navier-Stokes equations or a kineticequation like the lattice BGK equation) so that the total flow field(that is, the coarse eddies and the cascaded eddies), when filtered byan appropriate coarse-scale filter, yields a flow field that isstatistically similar to the flow field obtained in step 410. In anotherexample, a relaxation scheme may be used to relax (perhaps, withscale-dependent relaxation times/distances) the coarse-scale part of thefull flow towards the coarser eddy flow obtained in step 410. In stillother implementations, the coarse-scale eddies can be locked between thefield of step 410 and that of step 420.

In a more particular example of using a relaxation scheme withNavier-Stokes equations, the basic form of the Navier-Stokes equation

${\frac{\partial v}{\partial t} + {v \cdot {\nabla\; v}}} = {{- \frac{1}{\rho}}{\nabla_{p}{+ v}}{\nabla^{2}v}}$

may be modified to include additional terms representing extra forces,filters or relaxation terms. One example of such a modifiedNavier-Stokes equation that may be used to compute eddies in step 420 is

${\frac{\partial v}{\partial t} + {v \cdot {\nabla\; v}}} = {{{- \frac{1}{\rho}}{\nabla_{p}{+ v}}{\nabla^{2}v}} - \frac{v - u}{\sigma}}$

where u is the coarse-scale field computed in step 410 and σ is arelaxation time, which may be scale dependent.

In another example, the basic form of lattice BGK equation

${\frac{\partial f}{\partial t} + {v \cdot {\nabla\; f}}} = {--\frac{f - f_{eq}}{\tau}}$

can be modified to be used in step 420 computation of cascaded eddies

${\frac{\partial f}{\partial t} + {v \cdot {\nabla\; f}}} = {{--\frac{f - f_{eq}}{\tau}} - \frac{f - f_{LS}}{\sigma}}$

where f_(LS) is a (Large Scale) single-particle distribution functionthat generates the coarse-scale field u and σ is a relaxation term,which may be scale dependent. In typical implementations, the flowobtained in step 420 depends on Reynolds number and on grid resolution.In contrast to DNS (direct numerical simulation of turbulence), forexample, the method of step 420 can be designed so that there ismitigated dependence on grid resolution and Reynolds number (which arethe most significant restrictions of DNS or LES (coarse-eddysimulation)). These latter restrictions on DNS/LES include unphysicalbehavior if the grid resolution is too coarse for the given Reynoldsnumber—often seen as unphysically large energies at or near the gridscale. The key defect of DNS and LES that is avoided by the CEStechniques is the requirement that the turbulence dissipation scale bemuch coarser than the grid resolution, sometimes cited to be a factor of16 or more. With CES, the factor of 16 can be reduced to the order of 1or 2.

In addition, the spatial and temporal resolution in step 420 can bedifferent than in step 410. Furthermore, step 420 can have a differentMach number to account for accurate acoustic propagation and resonanceeffects. Step 410 need not account for the latter effects and may havethe Mach number be small or even zero (representing an incompressibleflow). Also, the computational domains for step 420 can be differentfrom step 410. The domain for step 420 may be a subdomain of that ofstep 410, which may help increase efficiency of the computations andreduce computer memory requirements. The domain for step 420 may be alarger domain than the domain of step 410, which includes more distantplaces due to its very low numerical dissipation. On the contrary,acoustic information in step 410 may be unable to propagate oversubstantial spatial distance due to turbulent eddy viscosity.

FIG. 5 shows an example process 500 for computing some coarse-scaleeddies of a turbulent flow. The example process 500 may be animplementation of step 410 described previously with respect to FIG. 4.The process includes accessing, for example from computer memory, aturbulent transport model (step 510) and applying the turbulenttransport model to generate a turbulent transport simulation ofcoarse-scale eddies (step 520). One implementation of a turbulenttransport model may be a turbulence model developed by supplementing arenormalization group approach with scale expansions for the Reynoldsstress and production of dissipation terms. As shown by arrow 530, theresults of the process 500 represent the coarse-scale eddies 210 in thereal world turbulent flow 200 described previously with respect to FIG.2.

FIG. 6 shows an example process 600 for computing cascaded eddiesresulting from the coarse-scale eddies of a turbulent flow. The exampleprocess 600 may be an implementation of step 420 described previouslywith respect to FIG. 4. The process 600 receives results 410R ofcoarse-scale eddy computation of the turbulent flow, such as the resultof step 410 described previously. The process 600 includes determininginitial forcing to be applied in the basic flow equations, such as theNavier-Stokes equation or the Lattice Boltzmann (LBE) equation (step610). In one example, as described previously, a relaxation term oranother type of damping term may be added to the basic flow equation todrive the result of the computation toward the results obtained for thecoarse-scale eddies. In another example, a driving term may be added toa basic flow equation to force the result computation toward the resultsobtained for the coarse-scale eddies.

After computing finer-scale eddies using the (modified) basic flowequations for a period of time (step 620), the partial computationresults are checked to determine whether the forcing being applied inthe modified basic equations needs to be changed (step 630). Stateddifferently, a modified force feedback loop may be applied to thecomputation of finer-scale eddies to ensure that the results of thecomputation of coarse-scale eddies are adequately reflected in theresults of computing the finer-scale eddies. If a determination is madethat the forcing applied by the basic flow equations needs to be changedto reflect the results of computing the coarse-scale eddies, the forcingto be applied is determined again (step 610) and computation continuesof finer-scale eddies continues using the modified basic flow equations(step 620). When required modifications of force is sufficiently small,the eddies at the finer-scale are deemed as having been computed and thecomputation of finer-scale eddies of the turbulent flow ends (step 640).

As shown by arrow 650, the results of the process 600 represent theintermediate-scale eddies 220 in the real world turbulent flow 200described previously with respect to FIG. 2. In addition, as shown byarrow 660, the results of the process 600 are used to provide moredetail for coarse-scale eddies 210 of the turbulent flow. Stateddifferently, the results of process 600 are used to help refine thesimulation of the coarse-scale eddies 210 of the turbulent flow byproviding greater granularity and better quality of flow data.

FIGS. 7A and 7B respectively illustrate the qualitative energy spectrums700A and 700B of a turbulent flow. As shown, spectrum 700A illustratesthe results of computation of coarse-scale eddies and finer-scale eddieswithout forcing the computation of finer-scale eddies, as describedpreviously with respect to FIG. 5. In contrast, spectrum 700B shows theresults of forcing the computation of finer-scale eddies, such asthrough applying a relaxation model or a dynamical constraint. Withoutforcing the computation of finer-scale eddies to stay within constraintsof the coarse-scale eddy, the flow computation of the finer-scale eddiesbreaks down and becomes physically inaccurate. For example, asillustrated in FIG. 7A, the finer eddies at low resolution appear tohave higher energy than the coarse scale eddies, which is not correctfrom a physical perspective.

Referring to FIG. 8, a system 800 can be used to implement thetechniques described above. As illustrated, the system 800 is a computersystem that includes a processor 810, a memory 820, a mass storagedevice 830, input/output devices 840A, 840B and 840C, and acommunication card or device 850. The system 800 also includes a systembus 860 that connects, directly or indirectly, each of the components810, 820, 830, 840A, 840B, 840C and 850.

The processor 810 is capable of processing instructions for executionwithin the system 800. The processor 810 may be a single-core,single-threaded processor or a multi-core and/or multi-threadedprocessor or a cluster of two or more such processors. The processor 810is capable of processing instructions stored in the memory 820 or on thestorage device 830.

The memory 820 is an internal data storage area of the computer 800. Thememory 820 may be implemented using computer chips capable of storingdata. The memory 820 may be referred to as main memory, core memory orphysical memory. The memory 820 may be volatile or non-volatile.

The mass storage device 830 is capable of providing mass storage for thesystem 800. The mass storage device 830 is a computer-readable medium.For example, the storage device may use any storage media (includingmagnetic, optical or solid state storage media) or any type of storagedevice (including a magnetic drive (such as a hard disk or a floppydisk), various types of compact discs (CDs) and DVDs (“digital videodiscs”), flash memory, or a solid-state device. In some implementations,the storage device 830 may be used to provide virtual memory for thecomputer system 800.

The input/output devices 840A-840C provides input/output operations forthe system 800. In particular, the computer system 800 includes adisplay device 840A, a keyboard 840B and a pointing device 840C.

The communication card or device 850 is operable to exchange data with anetwork 870 using a communication link 875 (such as a telephone line, awireless network link, a wired network link, or a cable network).Examples of a communication card or device 850 include a modem and anetwork adapter.

Other examples of system 800 are a handheld device, a workstation, aserver, a device, a component, other equipment, or some combination ofthese capable of responding to and executing instructions in a definedmanner. Any of the foregoing may be supplemented by, or incorporated in,ASICs (application-specific integrated circuits).

CES techniques are compatible with ‘out-of-core’ simulations that arequite challenging for other methods. In this case, the fully resolvedflow field at a given Reynolds number would require so much computermemory that it could not be stored within central memory (such ascomputer memory 820) but would, in addition, require substantialsecondary storage (such as mass storage device 830). More particularly,coarse-scale eddies of step 410 s may be computed ‘in-core’ (such asusing computer memory 820) without the use of secondary storage (such asmass storage device 830). Core or main computer memory generally wouldbe sufficient to compute the coarse-scale eddies because the resolutionrequirements of that computation (i.e., step 410) are not Reynoldsnumber dependent. The computation of finer-scale eddies, as in step 420,may require secondary storage, such as mass storage device 830.Moreover, the flow simulations computed for finer-scale eddies (as instep 420) are effectively driven (that is, forced) by the computation ofcoarse-scale eddies in step 410. As such, locally regular grids could beused, where such grids are more easily amenable to “out-of-core” memorysimulations because data structures used are typically simple. As anexample of this ‘out-of-core’ memory simulation, which also illustratesthe kind of simplifications possible in step 420, but possibly not incomplex geometry DNS or LES, is the use of ‘hyperviscosity’ or anothermethod to more effectively describe fine-scale dissipation than possiblewith Newtonian or simple sub-grid viscosities. The physical reason thatCES works in these latter cases, but DNS/LES does not, is that theturbulence generation mechanisms are effectively described in step 410but the Reynolds-number dependent dissipation scales only appear in step420 as eddies driven by the results of step 410. It is this carefulbalance between resolution and physics afforded by the minimally twostep CES process that is a key to some of the disclosed techniques.

Referring again to FIG. 4, the process 400 in some implementations maycompute additional cascaded eddies at an even finer scale using resultsof the finer-scale eddy computation of step 420 to constrain flowdynamics of the additional cascaded eddies (step 430). Step 430generally extends resolution limited scales to scales that are muchfiner than achievable by using a two-step flow simulation of steps 410and 420.

In some cases, especially at high Reynolds numbers, a three-step process(steps 410 to 430) may be employed to obtain the dissipation scale ofthe turbulence, which is typically of order of R times finer than thecoarse-scales of the turbulence. For example, if R is of order 10⁸ andtypical supercomputers can handle a range of scales of order 10³, thereare at least five orders of magnitude difference between the results ofthe two-step process (steps 410 and 420) and the results required forfull turbulence simulation. In most of the volume of high Reynoldsnumber flows, the additional range of scales may play a relativelysmaller role, but it is well known from turbulence theory thatturbulence is highly intermittent at fine scales in localized regions.The purpose of step 430 and beyond is to systemically allow thesimulation of these highly intermittent, yet important, flow regions.The basic idea is to do a conditional sample of the flow to chooserepresentative flow regions where significant activity may take place atfine scales (step 430A). If a random sampling of flow regions were madewithout conditioning this sampling, there may be little advantagecompared to a full high-resolution simulation at the high Reynoldsnumber, which is simply not feasible today. The conditional sampling canbe done, for example, by conditioning on local shear, local vorticity, acombination of shear and vorticity, or another flow characteristic. Oncethe local flow region is identified (and there may be many of them ateach subsequent step), the next step is to simulate the flow at thehighest feasible resolution in this local flow region (step 430B). Thiscan be done by various methods. In some implementations, the techniqueis to follow that of roughly step 420 in which the fine scale simulationinvolves relaxing or locking into the flow scales the coarsest scales ofthe fine scale simulation computed in the previous step of the procedureat the location of the fine scale simulation. Once this relaxation orlocking is accomplished, the flow Reynolds number at that simulationstep (here, step 420 or a subsequent step) is adjusted so that anaccurate simulation on the available grid is accomplished. Thisgenerates data at succeedingly finer scales, and describes theintermittent generation of localized turbulence structures. After anumber of steps of order log R, the full scale simulation at Reynoldsnumber R is accomplished. Generally, step 430 may be referred to as athird step or step 3 of the CES process.

FIG. 9 shows an example process 900 for extending computation to theturbulent flow of finer-scale eddies. The example process 900 may be animplementation of step 430 described previously with respect to FIG. 4.The process 900 uses results 420R of intermediate-scale eddy computationof the turbulent flow, such as the result of step 420 describedpreviously.

The process 900 includes identifying regions of interest (step 910), asdescribed previously with respect to step 430A of FIG. 4. The process900 also includes computing finer-scale eddies for each of theidentified regions (steps 920 and 930). In the example process 900, aprocess used to compute the intermediate-scale eddies in step 920 issubstantially similar to the process 600 described previously withrespect to FIG. 6. The process used to compute finer-scale eddies for anidentified region, however, need not necessarily be the same as theprocess used to compute eddies of another scale.

More particularly, computing finer-scale eddies for an identified regionincludes determining the initial forcing to be applied in the basic flowequations (step 920A), as described previously with respect to step 610of FIG. 6. Similarly to steps 620 and 630 of FIG. 6, after computingfiner-scale eddies using the (modified) basic flow equations for aperiod of time (step 920B), the partial computation results are checkedto determine whether the forcing being applied in the modified basicequations needs to be changed (step 920C). If a determination is madethat the forcing applied in the basic flow equations needs to be changedto reflect the results of computing the coarse-scale eddies, the forcingto be applied is determined (step 920A) and computation of finer-scaleeddies continues using the modified basic flow equations (step 920B).The computation of finer-scale eddies of the turbulent flow ends whenthe eddies at the finer-scale have been computed (step 920D).

Once computing finer-scale eddies at an identified region is completed(step 920), the process repeats by computing finer-scale eddies atanother identified region until finer-scale eddies have been computedfor all identified regions (step 930).

Alternatively or additionally, in lieu of identifying the regions ofinterest prior to computing finer-scale eddies for one region ofinterest (in step 910), a region of interest may be identified,finer-scale eddies may be computed for the identified region, adetermination may be made as to whether finer-scale eddies are to becomputed for another region, and, if so, the finer-scale eddies for theother region are computed, and so on until a determination is made thatthere are no more regions of interest for which finer-scale eddies areto be computed.

As shown by arrow 960, the results of the process 900 represent the fineinertial-scale eddies 230 in the real world turbulent flow 200 describedpreviously with respect to FIG. 2. In addition, as shown by arrow 970,the results of the process 900 are used to provide more detail forintermediate-scale eddies 220 of the turbulent flow. The results of theprocess 900 may be propagated to further refine the coarse-scale eddies,as illustrated by arrow 980. As such, a portion of the coarse-scaleeddies may be refined to show the indirect influence of the finer scaleeddies on the coarse-scale eddies.

The process of using results of a coarser-scale eddy computation toinfluence the computation of finer-scale eddies may be iterativelycontinued as necessary until dissipative scale eddies of the turbulentflow are computed.

The described systems, methods, and techniques may be implemented indigital electronic circuitry, computer hardware, firmware, software, orin combinations of these elements. Apparatus embodying these techniquesmay include appropriate input and output devices, a computer processor,and a computer program product tangibly embodied in a machine-readablestorage medium or device for execution by a programmable processor. Aprocess embodying these techniques may be performed by a programmableprocessor executing a program of instructions to perform desiredfunctions by operating on input data and generating appropriate output.The techniques may be implemented in one or more computer programs thatare executable on a programmable system including at least oneprogrammable processor coupled to receive data and instructions from,and to transmit data and instructions to, a data storage system, atleast one input device, and at least one output device. Each computerprogram may be implemented in a high-level procedural or object-orientedprogramming language, or in assembly or machine language if desired; andin any case, the language may be a compiled or interpreted language.Suitable processors include, by way of example, both general and specialpurpose microprocessors. Generally, a processor will receiveinstructions and data from a read-only memory and/or a random accessmemory. Storage devices suitable for tangibly embodying computer programinstructions and data include all forms of non-volatile memory,including by way of example semiconductor memory devices, such asErasable Programmable Read-Only Memory (EPROM), Electrically ErasableProgrammable Read-Only Memory (EEPROM), and flash memory devices;magnetic disks such as internal hard disks and removable disks;magneto-optical disks; and Compact Disc Read-Only Memory (CD-ROM). Anyof the foregoing may be supplemented by, or incorporated in,specially-designed ASICs (application-specific integrated circuits).

It will be understood that various modifications may be made withoutdeparting from spirit and scope of the claims. For example, the steps ofthe disclosed techniques and concepts may be performed in a differentorder and/or the components in the disclosed systems may be combined ina different manner and/or replaced or supplemented by other components.Other implementations are within the scope of the following claims.

1-32. (canceled)
 33. A computer program for simulating a fluid flow, thecomputer program causing a computer to: perform operations on datastored in the memory to compute a first set of results that include atleast one coarse eddy of a transient fluid flow; and determine amodification of a flow equation used to compute at least one cascadededdy of the fluid flow, the modification including one or moreadditional terms representing additional forces, filters, dampeningterms, or relaxation terms; and perform operations to supplement thefirst set of results that includes the coarse eddy of the transient flowby computing a second set of results that include at least one cascadededdy of the fluid flow using the flow equation as modified such that-thecomputation of the second set of results that includes the at least onecascaded eddy of the fluid flow is constrained by the first set ofresults of the computation of the at least one coarse eddy of the fluidflow.
 34. The computer program of claim 33 wherein the computation ofthe at least one cascaded eddy of the fluid flow is constrained byforcing the computation of the second set of results by applying arelaxation model or a dynamical constraint such that an energy spectrumof the fluid flow is substantially the same statistically as the resultsof the computation of the at least one coarse eddy of the fluid flow.35. The computer program of claim 33 further configured to cause thecomputer to perform operations to compute a third set of results thatinclude at least one additional cascaded eddy of the fluid flow wherein:the computation of the at least one additional cascaded eddy of thefluid flow is constrained by results of the computation of the at leastone cascaded eddy of the fluid flow.
 36. The computer program of claim35 further configured to: identify a region of the fluid flowrepresenting a region of activity that is significant to the computingof the fluid flow; and perform operations to compute at least oneadditional cascaded eddy of the identified region of the fluid flow. 37.The computer program of claim 36 further configured to performoperations to conditionally sample the fluid flow to identify the regionof significant activity.
 38. The computer program of claim 36 whereinconditionally sampling the fluid flow comprises conditionally samplingat least one of local shear, local vorticity and a combination of shearand vorticity.
 39. The computer program of claim 33 wherein performingoperations to compute at least one cascaded eddy of the fluid flowfurther comprises: determining whether the forcing being applied in themodified flow equations is to be modified, in response to adetermination that the forcing is to be modified, determine a secondforcing to be applied by a flow equation used to compute the at leastone cascaded eddy of the fluid flow, and computing the at least one eddyof the fluid flow using the flow equation modified by the determinedsecond forcing.
 40. The computer program of claim 33 wherein themodified flow equation used to compute the at least one cascaded eddy ofthe fluid flow comprises a modified Navier-Stokes equation.
 41. Thecomputer program of claim 33 wherein the modified flow equation used tocompute the at least one cascaded eddy of the fluid flow comprises amodified kinetic equation.
 42. The computer program of claim 33 whereinthe modified flow equation used to compute the at least one cascadededdy of the fluid flow comprises a modified flow equation that includesa relaxation term.
 43. The computer program of claim 33 wherein themodified flow equation used to compute the at least one cascaded eddy ofthe fluid flow comprises a modified flow equation that includes adampening term.
 44. A computer system for simulating a fluid flow, thesystem comprising a processor, a memory and a mass storage device andbeing configured to: perform operations on data stored in the memory tocompute a first set of results that include at least one coarse eddy ofa transient fluid flow; and determine a modification of a flow equationused to compute at least one cascaded eddy of the fluid flow, themodification including one or more additional terms representingadditional forces, filters, dampening terms, or relaxation terms; andperform operations to supplement the first set of results that includesthe coarse eddy of the transient flow by computing a second set ofresults that include at least one cascaded eddy of the fluid flow usingthe flow equation as modified such that-the computation of the secondset of results that includes the at least one cascaded eddy of the fluidflow is constrained by the first set of results of the computation ofthe at least one coarse eddy of the fluid flow.
 45. The computer systemof claim 44 wherein the computation of the at least one cascaded eddy ofthe fluid flow is constrained by forcing the computation of the secondset of results by applying a relaxation model or a dynamical constraintsuch that an energy spectrum of the fluid flow is substantially the samestatistically as the results of the computation of the at least onecoarse eddy of the fluid flow.
 46. The computer system of claim 44further configured to perform operations to compute a third set ofresults that include at least one additional cascaded eddy of the fluidflow wherein: the computation of the at least one additional cascadededdy of the fluid flow is constrained by results of the computation ofthe at least one cascaded eddy of the fluid flow.
 47. The computersystem of claim 46 further configured to: identify a region of the fluidflow representing a region of activity that is significant to thecomputing of the fluid flow; and perform operations to compute at leastone additional cascaded eddy of the identified region of the fluid flow.48. The computer system of claim 47 further configured to performoperations to conditionally sample the fluid flow to identify the regionof significant activity.
 49. The computer system of claim 47 whereinconditionally sampling the fluid flow comprises conditionally samplingat least one of local shear, local vorticity and a combination of shearand vorticity.
 50. The computer system of claim 44 wherein performingoperations to compute at least one cascaded eddy of the fluid flowfurther comprises: determining whether the forcing being applied in themodified flow equations is to be modified, in response to adetermination that the forcing is to be modified, determine a secondforcing to be applied by a flow equation used to compute the at leastone cascaded eddy of the fluid flow, and computing the at least one eddyof the fluid flow using the flow equation modified by the determinedsecond forcing.
 51. The computer system of claim 44 wherein the modifiedflow equation used to compute the at least one cascaded eddy of thefluid flow comprises a modified Navier-Stokes equation.
 52. The computersystem of claim 44 wherein the modified flow equation used to computethe at least one cascaded eddy of the fluid flow comprises a modifiedkinetic equation.
 53. The computer system of claim 44 wherein themodified flow equation used to compute the at least one cascaded eddy ofthe fluid flow comprises a modified flow equation that includes arelaxation term.
 54. The computer system of claim 44 wherein themodified flow equation used to compute the at least one cascaded eddy ofthe fluid flow comprises a modified flow equation that includes adampening term.